Question: What is the remainder when 1,234,567,890 is divided by 99?
We can write 1234567890 as \[12 \cdot 10^8 + 34 \cdot 10^6 + 56 \cdot 10^4 + 78 \cdot 10^2 + 90.\] Note that \[10^8 - 1 = 99999999 = 99 \cdot 1010101,\] is divisible by 99, so $12 \cdot 10^8 - 12$ is divisible by 99.

Similarly, \begin{align*}
10^6 - 1 &= 999999 = 99 \cdot 10101, \\
10^4 - 1 &= 9999 = 99 \cdot 101, \\
10^2 - 1 &= 99 = 99 \cdot 1
\end{align*} are also divisible by 99, so $34 \cdot 10^6 - 34$, $56 \cdot 10^4 - 56$, and $78 \cdot 10^2 - 78$ are all divisible by 99.

Therefore, \[12 \cdot 10^8 + 34 \cdot 10^6 + 56 \cdot 10^4 + 78 \cdot 10^2 + 90 - (12 + 34 + 56 + 78 + 90)\] is divisible by 99, which means that $1234567890$ and $12 + 34 + 56 + 78 + 90$ leave the same remainder when divided by 99.

Since $12 + 34 + 56 + 78 + 90 = 270 = 2 \cdot 99 + 72$, the remainder is $\boxed{72}$.